Consider a well-reservoir system (Fig. 1) consisting of:
- producing well W1 with total sandface flowrate
q_1(t)>0 and BHP
p_1(t)>0, draining the reservoir volume
V_{\phi, 1}
water injecting well W0 with total sandface flowrate q_0(t) <0, supporting pressure in reservoir volume V_{\phi, 0}
The injection drainage volume
V_{\phi, 0} includes the drainage volume
V_{\phi, 1} of producer W1 and may be equal to it
V_{\phi, 0} = V_{\phi, 1} or may be bigger
V_{\phi, 0} > V_{\phi, 1} in case injector W0 supports other producers {W1 .. WN}:
V_{\phi, 0} = \sum_{k=1}^N V_{\phi, k}.
Fig. 1. Location map of injector-producer pairing with 4 producers {W1, W2, W3, W4} and one injector W0. |
Case #1 – Constant flowrate production: q_1 = \rm const >0
The bottom-hole pressure response \delta p_1 in producer W1 to the flowrate variation \delta q_0 in injector W0:
(1) | \delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_2 |
where
t | time since the water injection rate has changed by the \delta q_2 value. |
p_{u,\rm 01}(t) | cross-well pressure transient response in producer W1 to the unit-rate production in injector W0 |
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Case #2 – Constant BHP: p_1 = \rm const
Assume that the flowrate in producer W1 is being automatically adjusted by \delta q_1(t) to compensate the bottom-hole pressure variation \delta p_1(t) in response to the total sandface flowrate variation \delta q_2 in injector W0 so that bottom-hole pressure in producer W1 stays constant at all times \delta p_1(t) = \delta p_1 = \rm const. In petroleum practice this happens when the formation is capable to deliver more fluid than the current lift settings in producer so that the bottom-hole pressure in producer is constantly kept at minimum value defined by the lift design..
In this case, flowrate response \delta q_1 in producer W1 to the flowrate variation \delta q_0 in injector W0 is going to be:
(4) | \delta q_1(t) = - \frac{\dot p_{u,\rm 01}(t)}{\dot p_{u,\rm 11}(t)} \cdot \delta q_0 |
where
t | time since injector's W0 rate has changed by \delta q_0. |
\dot p_{u,\rm 01}(t) | time derivative of cross-well pressure transient response (CTR) in producer W1 to the unit-rate production in injector W0 |
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\dot p_{u,\rm 11}(t) | time derivative of drawdown pressure transient response (DTR) in producer W1 to the unit-rate production in the same well |
For the finite-volume drain V_{\phi,1} \leq V_{\phi,0} < \infty the flowrate response factor \delta q_1 / \delta q_0 is getting stabilised over time as:
(11) | \delta q_1 / \delta q_0 = - f_{01} = - \frac{V_{\phi, 1}}{ V_{\phi, 0}} = \rm const |
The response delay in time still exists but in usual time-scales of production analysis it becomes negligible and one can consider
(11) as constant in time.
In case injector W0 supports only one producer W1 , then both wells drain the same reservoir volume
V_{\phi, 0} = V_{\phi, 1} so that
(11) leads to:
(14) | \delta q_1 = -\delta q_0 |
which means that producer W1 with constant BHP and finite-reservoir volume will eventually vary its rate at the same volume as injector W0.
In case injector W0 supports many producers {W1 .. WN} then all injection shares towards producers are going to sum up to a unit value:
(15) | \sum_{k=1}^N f_{0k} = 1 |
with constant coefficients f_{0k} \geq 0, \ {k=\{i..N \} }, unless there is a thief injection outside the drainage area of all producers and in this case:
(16) | \sum_{k=1}^N f_{0k} < 1 |
If pressure around producer W1 is supported by several injectors
N_{\rm inj} > 1 then one can assume:
(17) | \delta q_1 =\sum_k f_{i1} \delta q_i |
with constant coefficients f_{i1} \geq 0, \ {k=\{i..N_{\rm inj} \} }.
The equations (15), (16) and (17) make one of the key assumptions in Capacitance Resistance Model (CRM).
It is important to note that assumption that injector W0 may drain bigger volume than producer W1 V_{\phi, 0}> V_{\phi, 1} is a misnomer.
When wells (producers and injectors) are placed into interconnected reservoir volume they drain the same volume V_\phi alltogether and the DTR/CTR will have the same LTR asymptotic:
(18) | p_{u,\rm ik}(t \rightarrow \infty ) \rightarrow \frac{t}{c_t \, V_\phi}, \forall i,k. |
Moreover, if each well is placed in different reservoir volumes which are only connected through wellbores then again they will all drain the same volume which is the sum of all connected volumes through the wellbores and the DTR/CTR will again trend to the same LTR asymptotic.
In order to relate the DTR/CTR from numerical grid simulations or from deconvolution theory to the CRM injection share constants f_{ik} one need to implement a following trick.
See also
[ DTR ] [ CTR ] [ Capacitance Resistance Model (CRM) ]